Abstract : Partial differential equations are a family of powerful mathematical tools to model the physical world. In many practical situations, it is generally very difficult to obtain their analytical solutions, and thus numerical methods become critical for simulation. This mini-symposium aims to bring together scholars to discuss the recent advances and innovative techniques in this field including finite element exterior calculus, physics-preserving schemes, polytopal meshes and high-order methods. The mini-symposium will also deal with the applications onto electromagnetism, fluid dynamics and interface problems.
Organizer(s) : Long Chen, Ruchi Guo, Liuqiang Zhong
[03945] Geometrical degrees of freedom for high order Whitney forms
Format : Talk at Waseda University
Author(s) :
Ana María Alonso Rodríguez (University of Trento)
Francesca Rapetti (Université Côte d’Azur)
Ludovico Bruni Bruno (Università dell'Insumbria)
Abstract : Finite element spaces extending Whitney forms to higher polynomial degrees are widely used for discretizing partial differential equations in electromagnetism, fluid dynamics or elasticity, and different degrees of freedom (dofs) can be considered for interpolation. In particular the so-called weights preserve the meaning of the natural degrees of freedom associated with Whitney forms as circulation, fluxes or densities since they are the integrals of a k-form on k-chains. Weights are a generalization of the evaluations of a scalar function at a set of nodes in view of its reconstruction on multivariate polynomial bases and allows to extend in a natural way well- known concepts as the Lebesgue constant. We rely on the flexibility of the weights with respect to their geometrical support to reduce the growth of the Lebesgue constant when increasing the degree of the polynomial interpolation of differential k-forms.
[02919] Energy-preserving Mixed finite element methods for a ferrofluid flow model
Format : Talk at Waseda University
Author(s) :
Yongke Wu (University of Electronic Science and Technology of China)
Xiaoping Xie (Sichuan University)
Abstract : In this talk, we introduce a class of mixed finite element methods for the ferrofluid flow model proposed by Shliomis [Soviet Physics JETP, 1972]. We show that the energy stability of the weak solutions to the model is preserved exactly for both the semi- and fully discrete finite element solutions. Furthermore, we prove the existence of the discrete solutions and derive optimal error estimates for both the the semi- and fully discrete schemes. Numerical experiments confirm the theoretical results.
[03132] Immersed CR element methods for the elliptic and Stokes interface problems
Format : Talk at Waseda University
Author(s) :
Haifeng Ji (Nanjing University of Posts and Telecommunications)
FENG WANG (NanJing Normal University)
Jinru Chen (NanJing Normal University)
Zhilin Li (North Carolina State University)
Abstract : In this talk, we shall discuss the immersed CR element method for solving the elliptic and Stokes interface problems with piecewise constant coefficients that have jumps across the interface. In the method, the triangulation does not need to fit the interface and the IFE spaces are constructed from the traditional CR element with modifications near the interface according to the interface jump conditions. The stability and the optimal error estimates of the proposed methods are also derived rigorously. The constants in the error estimates are shown to be independent of the interface location relative to the triangulation. Numerical examples are provided to verify the theoretical results.
[02917] A fast Cartesian grid method for unbounded interface problems with non-homogeneous source terms
Format : Talk at Waseda University
Author(s) :
Jiahe Yang (Shanghai Jiao Tong University)
Wenjun Ying (Shanghai Jiao Tong University)
Abstract : A Cartesian grid method is presented for interface problems of PDEs with non-homogeneous source terms on unbounded domains. The method adapts a compression-decompression technique and has algorithm complexity of only $O(n^2\log n)$ as compared to $O(n^3)$ by traditional methods. This Cartesian grid method is an extension of the kernel-free boundary integral method, which avoids direct evaluation of singular or nearly singular integrals by reformulating them into solutions of equivalent but much simpler interface problems.
[04905] Staggered DG methods for elliptic problems on general meshes
Format : Talk at Waseda University
Author(s) :
Eun-Jae Park (Yonsei University)
Abstract : In this talk, we present our recent framework on staggered DG
methods for elliptic equations on general meshes, which can be
flexibly applied to rough grids such as highly distorted meshes.
Adaptive mesh refinement is an attractive tool for general meshes
due to their flexibility and simplicity in handling hanging nodes.
We derive a simple residual-type error estimator. Numerical
results indicate that optimal convergence can be achieved for both
the potential and vector variables, and the singularity can be
well-captured by the proposed error estimator. Then, some
applications to Darcy-Forchheimer equations, Stokes equations, and
linear elasticity equations are considered. This is joint work
with Eric Chung, Dohyun Kim, Dong-wook Shin, and Lina Zhao.
[04160] Accelerated Gradient and Skew-Symmetric Splitting Methods for Monotone Operator Equations
Format : Online Talk on Zoom
Author(s) :
Jingrong Wei (University of California, Irvine)
Long Chen (University of California at Irvine)
Abstract : A class of monotone operator equations, which can be decomposed into sum of a gradient of a strongly convex function and a linear and skew-symmetric operator, is considered in this work. Based on discretization of the generalized accelerated gradient flow, accelerated gradient and skew-symmetric splitting (AGSS) methods are developed and shown to achieve linear rates with optimal lower iteration complexity when applied to smooth saddle point systems with bilinear coupling.
[03693] Solve electromagnetic interface problems on unfitted meshes
Format : Talk at Waseda University
Author(s) :
Ruchi Guo (University of California Irvine)
Abstract : Electromagnetic interface problems widely appear in a lot of engineering applications, such as electric actuators, invasive detection techniques and integrated circuit，which are typically described by Maxwell equations with discontinuous coefficients. Conventional finite element methods require a body-fitted mesh to solve interface problems, but generating a high-quality mesh for complex interface geometry is usually very expensive. Instead using unfitted mesh finite element methods can circumvent mesh generation procedure, which greatly improve the computational efficiency. However, the low regularity of Maxwell equations makes its computation very senstive to the conformity of the approximation spaces. This very property poses challenges on unfitted mesh finite element methods, as most of them resort to non-conforming spaces. In this talk, we will present our recent progress including several methods for this topic.
[03714] Pressure-robust virtual element methods for the Stokes problem on polygonal meshes
Format : Talk at Waseda University
Author(s) :
Gang Wang (Northwestern Polytechnical University)
Abstract : In this talk, we shall introduce two pressure-robust virtual element methods for the Stokes problem on polygonal meshes. The standard virtual element scheme involves a pressure contribution in the velocity error. To achieve the pressure-independent velocity approximation, we define an H(div)-conforming velocity reconstruction operator for the velocity test function and propose the modified scheme by employing it in the approximation of right-hand-side source term assembling. In the first method, we apply the H(div)-conforming elements on the polygons and construct a (theoretically) exactly pressure-robust virtual element scheme. Since basis functions of H(div)-conforming elements are not polynomials on general polygons, quadrature errors will affect the exact pressure-robustness in implementation. To solve it, we reconstruct a (theoretically and numerically) exactly pressure-robust virtual element scheme by designing H(div)-conforming element based on the sub-triangulation of polygon and the lowest-order Raviart-Thomas element. We give the error estimates of two methods and also show numerical examples to support our theories.
[05142] Arbitrary order DG-DGLM method for hyperbolic systems of multi-dimensional conservation laws
Format : Talk at Waseda University
Author(s) :
Mi-Young Kim (Inha University)
Abstract : An arbitrary order discontinuous Galerkin method with Lagrange multiplier in space and time is proposed to approximate the solution to hyperbolic systems of multi-dimensional conservation laws. Weak formulation is derived through the definition of weak divergence. Weak solution on the edge is characterized as the average of the solutions on the elements sharing the edge. Stability of the approximate solution is proved in a broken $L_2(L_2)$ norm. Error estimates of $O(h^r + k_n^q)$ with $P_r(E)$ and $P_q(J_n)$ elements $(r, q > 1 + d/2)$ are then derived in a broken $L_2(L_2)$ norm, where $h$ and $k_n$ are the maximum diameters of the elements and the time step of $J_n,$ respectively, $J_n$ is the time interval, and $d$ is the dimension of the spatial domain. Some numerical examples are presented.
[04537] Implementation and Application of Virtual Element Method in FEALPy
Format : Talk at Waseda University
Author(s) :
Huayi Wei
HUAYI WEI (Xiangtan University )
Abstract : The virtual element method is a novel numerical solution technique for PDEs. It can be considered an extension of the finite element method to polygonal or polyhedral meshes. Due to its novelty, the program implementation of this method differs significantly from that of the traditional finite element method. Unfortunately, there are relatively few open-source program implementations available. FEALPy is an open-source numerical solution algorithm library for PDEs. It is built entirely on Python's basic scientific computing module and provides rich mesh data structures, meshes adaptive algorithms, and partial differential equation numerical discrete algorithms. This report primarily focuses on the design and implementation of the virtual element method in FEALPy, along with several typical application examples.
[05586] Convergence of an AWG method for indefinite time-harmonic Maxwell equations
Format : Talk at Waseda University
Author(s) :
Yingying Xie (Guangzhou University)
Liuqiang Zhong (South China Normal University)
Ming Tang (South China Normal University)
Abstract : In this talk, an adaptive weak Galerkin (AWG) method for indefinite time-harmonic Maxwell equations is studied. Firstly, a residual type a posteriori error estimator is presented and analyzed. Then, a quasi-orthogonality is presented by introducing interpolator operators. And the convergence of AWG algorithm is also proved. Finally, some numerical experiments is provided to support the theoretical results.
[04955] High order stable generalized finite element method for interface problems
Format : Talk at Waseda University
Author(s) :
Qinghui Zhang (Harbin Institute of Technology, Shenzhen)
Abstract : Generalized or Extended Finite Element Methods (GFEM/XFEM) of degree 1 (linear elements) for interface problems have been reported in the literature; they (i) yield optimal order of convergence in energy norm, i.e., O(h), (ii) are stable in a sense that conditioning is not worse than that of the standard FEM, and (iii) are robust in that the conditioning does not deteriorate as interface curves are close to boundaries of underlying elements. However, higher order GFEM/XFEM with the properties (i)-(iii) have not been successfully addressed yet. Various enrichment schemes for GFEM/XFEM based on D or DP_k (D is a distance function or the absolute value of level set function, and P_k is the polynomial basis of degree k) have been reported to obtain higher order convergence, but they are not stable or robust in general; in fact, they even may not yield the optimal orders of convergence. In this talk, we propose a stable GFEM/XFEM of degree 2 (SGFEM2) for the interface problems, where we use the enrichment scheme based on D{1, x, y}, instead of D or D{1,x, y, x^2, xy, y^2} in the literature. We prove that the SGFEM2 yields the optimal order of convergence, i.e., O(h^2), for the interface problems with curved (smooth) interfaces. A local principal component analysis technique is proposed, which ensures that the SGFEM2 is stable and robust. Numerical experiments for straight and curved interfaces have been presented to illuminate these properties.
